Wow, so none of the players I use have been mentioned at all...
PC
-Video
Zoom Player: The amount of features and user setting customizations is through the roof. You can resync audio with a keyboard shortcut, you can go frame-by-frame with audio with a keyboard shortcut (which is extremely helpful in syncing up audio), you can change how far/short the video jump in seconds when hitting left/right arrows, you can change how much volume increases/decreases with the up/down arrows, you can even have the mouse scroll wheel shrink/enlarge the player window 1 pixel at a time if you so desire.
-Audio
jetAudio: This player has hands-down the best sound quality with its BBE sound enhancements along with a 20-band equalizer. The interface takes some time to get used to as adding songs to its library requires some experimentation on the user's end to figure it out.
Mobile
-Video
MoboPlayer: This player has a slew of features for a mobile video player including stuff like subtitles to a floating window to keep a video playing while switching apps to say your Internet browser. You would think with all these features that an app like this would be full of ads with paying a few bucks for the premium version but MoboPlayer is completely ad-free. The developer even frequently updates the app, occasionally pushing out really fast updates to fix something the last update broke.
-Audio
jetAudio: Again, this player has hands-down the best sound quality out there. However, unlike its PC counterpart, it does not feature BBE sound enhancements. But, with that said, it does have other sound enhancements including the Crystalizer and Bongiovi DPS along with standard jetAudio effects (like bass, wide, etc.). It does take quite a bit of playing around with everything to get the sound just right as opposed to the much better ease of the BBE effects on the desktop version. The player features a 20-band equalizer as well and having the extra bands really makes a difference how much you can tailor the sound to your preferences. The player is full featured with lots of settings options along with the ability to download album art. Where the mobile version definitely outdoes the desktop version is interface, which is so much easier and better than the desktop version. The interface may not be the slickest but it's definitely super easy and extremely functional allowing you to view by your music by song, artist, album, and by folders. To get the most out of jetAudio, you definitely have to buy it to get the 20-band EQ and you have to buy the Crystalizer and Bongiovi enhancements (ignore the AM3D one), which I think costs about $10 in total. There's of course a free version to check out and you can enable the premium effects for like 10 seconds or so to check them out.
Here's my settings for jetAudio:
Equalizer: 1,-4,-2,-1,0,1,2,3,4,3,2,1,0,-1,-2,-3,-4,-3,-2,-2
jetAudio sound effects: X-Bass 2 at 18, X-Wide 2 at 32
Crystalizer:8
Bongiovi DPS set to 'New Orleans General'